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Let $(X,d)$ be a metric space and $(K_X , h_d)$ be the associated metric space of nonempty compact subsets of $X$ with the Hausdorff metric. It is well known that $K_X$ inherits certain topological (and analytic) properties from $X$. For example, if $X$ is compact, then so is $K_X$; and if $X$ is complete, then so is $K_X$.

Is there a reference which further explores the properties that $K_X$ inherits from $X$? In particular, if $X$ is locally compact then is $K_X$ also?

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  • $\begingroup$ "The associated metric space of compact subsets of 𝑋 with the Hausdorff metric": you mean of nonempty compact subsets. Otherwise I don't see how you measure the distance to the empty set. For $X$ is locally compact, you can indeed make the set of compact subsets (including the empty set) topological in a natural way, and $\emptyset$ is isolated in this space iff $X$ is compact; in all cases the space of nonempty compact subsets is locally compact. $\endgroup$
    – YCor
    Jun 22 '20 at 19:12
  • $\begingroup$ Yes, nonempty. I'll update the prompt $\endgroup$
    – Logan Fox
    Jun 22 '20 at 19:58
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    $\begingroup$ this is the reference, especially section $4$, showing that many properties of $X$ are preserved in $K(X)$ including being locally compact. Another one not mentioned in the paper is being zero dimensional. Regarding the empty set issue you can just set the distance between an empty set and a nonempty one to be $1$ and use the Hausdorff metric otherwise to get a metric compatible with the Vietoris topology on $K(X)$ $\endgroup$ Jun 22 '20 at 21:26
  • $\begingroup$ (I was assuming the metric of $X$ to be bounded by $1$ in the above comment concerning the empty set) $\endgroup$ Jun 22 '20 at 21:36
  • $\begingroup$ Check out books Illanes, Nadler - Hyperspaces and Beer - Topologies on closed and closed convex sets. Also, some basic useful facts can be found in Engelking in exercises $\endgroup$
    – erz
    Jun 22 '20 at 21:50
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Michael, Ernest. “Topologies on spaces of subsets.” Transactions of the American Mathematical Society 71 (1951): 152-182.

The above paper shows in section $4$ that many properties of $X$, including being locally compact, are inherited by $K(X)$ (the latter space is called $\mathcal C(X)$ in the paper, while the author uses $2^X$ to refer to the hyperspace of closed sets of $X$, for which a more modern notation is $F(X)$).

In my comment I said that being zero-dimensional is also preserved but not shown in the paper, I now noticed that it is in fact shown there, together with more connectedness properties. The relationship between $\dim X$ and $\dim K(X)$ is much harder in general and has been widely studied, there is a whole chapter concerning this topic in the book Hyperspaces by Illanes and Nadler.

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